An `alpha` particle and a proton have their masses in the ratio 4:1 and charges in the ratio 2: 1. Find ratio of their de-Broglie wavelengths when both move with equal velocities.

Options

A. `2:1`
B. `1:1`
C. `4:1`
D. `1:4`

Answer

To find the ratio of the de-Broglie wavelengths of an alpha particle and a proton when both move with equal velocities, we can follow these steps:

### Step 1: Understand the de-Broglie wavelength formula
The de-Broglie wavelength (λ) is given by the formula:
\[ \lambda = \frac{h}{mv} \]
where \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is its velocity.

### Step 2: Define the variables for the alpha particle and proton
Let:
- For the alpha particle (particle 1):
- Mass = \( m_1 \)
- Velocity = \( v_1 \)
- de-Broglie wavelength = \( \lambda_1 \)

- For the proton (particle 2):
- Mass = \( m_2 \)
- Velocity = \( v_2 \)
- de-Broglie wavelength = \( \lambda_2 \)

### Step 3: Write the expressions for the de-Broglie wavelengths
Using the formula for de-Broglie wavelength:
\[ \lambda_1 = \frac{h}{m_1 v_1} \]
\[ \lambda_2 = \frac{h}{m_2 v_2} \]

### Step 4: Find the ratio of the de-Broglie wavelengths
The ratio of the de-Broglie wavelengths is:
\[ \frac{\lambda_1}{\lambda_2} = \frac{h/m_1 v_1}{h/m_2 v_2} = \frac{m_2 v_2}{m_1 v_1} \]
Since both particles are moving with equal velocities, we have \( v_1 = v_2 \). Thus, we can simplify the ratio:
\[ \frac{\lambda_1}{\lambda_2} = \frac{m_2}{m_1} \]

### Step 5: Use the given mass ratio
According to the problem, the mass ratio of the alpha particle to the proton is given as:
\[ \frac{m_1}{m_2} = \frac{4}{1} \]
This means:
\[ m_1 = 4m_2 \]
Thus, we can write:
\[ \frac{m_2}{m_1} = \frac{1}{4} \]

### Step 6: Conclude the ratio of the de-Broglie wavelengths
Substituting this into the ratio of the de-Broglie wavelengths:
\[ \frac{\lambda_1}{\lambda_2} = \frac{m_2}{m_1} = \frac{1}{4} \]

### Final Result
Therefore, the ratio of the de-Broglie wavelengths of the alpha particle to the proton is:
\[ \lambda_1 : \lambda_2 = 1 : 4 \]